What is Bell's theorem?

I recently read about Bell's theorem in the book 'The Dancing Wu Li Masters' by Gary Zukav, and was fairly intrigued by its implications. Can anyone tell me of what the theorem actually consists? Zukav never actually put the mathematics forward, but I wanted to see if anyone could tell me what they were.

The following explantion is based on the one in Quantum Mechanics by Alastair Rae published by the IoP, though I've added a few bits and taken out others and generally changed the wording:

Consider a local hidden variable theory (LHVT) and a system of spin half atoms, as it's a LHVT the result of the measuremnt of the spins will be pre-detirmined before the measuremnt takes place.

Now consider the components of the spin in three directions (1,2,3). a set of N atoms will contain a subset of n(1+,2+,3+) particles, each of them would mean apostive result if there spin was measured in any of the three directions, wheras, particles that are members of the n(1-,2+,3+) would yield a negative result in the 1 direction and a postive in the 2 and 3 directions, etc., therefore the set of N particles conatins 8 mutually exclusive subsets as defined by their spin components in the three directions. You cannot define which atom is in which subset as measuring one spin component would change the others but as this is an LHVT all atoms must belong to one of the subsets.

Now lets say that each atom in the subset N is a member of an entangled pair (though obviously as we've assumed a LHVT they won't be truly entangled but they're measuremnts are still dependent on each other though pre-detirmined). By measuring the spin component of 1 member of the entangled pair in say the 1 direction and the other member in the 2 direction we will be able to know the components of spin in both the 1 and 2 directions of a particle by only disturbing it once as the measurements do not affect each other. we can by measuring these two components create 5 new subsets:

n(1+,2+) = n(1+,2+,3+) + n(1+,2+,3-)

n(1+,3+) = n(1+,2+,3+) + n(1+,2-,3+)

n(2-,3+) = n(1+,2-,3+) + n(1-,2-,3+)

etc.

If N is large enough we should be able to effectively measure any of the above three sets, also from the above we can detirmine:

Putting in the QM predictions for this (where &theta;12 is the anle between 1 and 2 obtained from the first equation):

n(1+,2+) =NP+-(&theta;12)

cos2 &theta;12/2 - cos2 &theta;13/2 + sin2 &theta;23/3 >= 0

Now lets say that all three measuremnt directions are in the same plane (as they are allowed to be), therefore:

&theta;12+&theta;23 = &theta;13, now specializing further and conmsidering the case when &theta;13 = 3&theta;12 and now taking &theta;12/2 as &theta; we get:

cos2&theta; + sin22&theta;¸ - cos23&theta; >= 0

Now we can then plot this function we find that when &theta; = 20°, among others, the value is actually negative (-0.22) therefore the last equation cannot be true and the predictions of a LHVT differs from those of QM and both cannot be true.

The other thing I was unsure about was that Bell's theorem supposedly proved somewhat that superluminal communication was possible at not only a microscopic but also a macroscopic level. Maybe I'm misinterpreting what Zukav was saying, but if someone could either confirm this interpretation, or explain what part I'm missing, I'd be very appreciative.

The other thing I was unsure about was that Bell's theorem supposedly proved somewhat that superluminal communication was possible at not only a microscopic but also a macroscopic level. Maybe I'm misinterpreting what Zukav was saying, but if someone could either confirm this interpretation, or explain what part I'm missing, I'd be very appreciative.

No quantum entanglement does not allow information be sent superluminally, but it does mean that you can by measuring the stae of one entangled particle detrimine the state of the other entangled particle an arbitarily large distance apart.

Zukav wrote something about how if you extend the mathematics a bit, and extrapolate on the ideas posed by Bell's theorem, it shows that the particles are communicating with other particles, and they know what the spin of its partner is. The example given is the experiment where there are two particles created, each with an opposite spin (in terms of left vs. right or up vs. down) and they head in opposite directions. One particle is put through a magnetic field and its spin is determined as being, up, down, left or right. Zukav says that instantly, we know that the other particle's spin is because of what the test shows. This is where it gets a bit iffy for me. He says something about how the particles have communicated superluminally either between themselves, or with the experimenters (i think he implies that option) and discovered what their spins should be.

Thats my understanding of it and I dont think I'm understanding what he's trying to say entirely.

Originally posted by cytokinesis Zukav wrote something about how if you extend the mathematics a bit, and extrapolate on the ideas posed by Bell's theorem, it shows that the particles are communicating with other particles, and they know what the spin of its partner is. The example given is the experiment where there are two particles created, each with an opposite spin (in terms of left vs. right or up vs. down) and they head in opposite directions. One particle is put through a magnetic field and its spin is determined as being, up, down, left or right. Zukav says that instantly, we know that the other particle's spin is because of what the test shows. This is where it gets a bit iffy for me. He says something about how the particles have communicated superluminally either between themselves, or with the experimenters (i think he implies that option) and discovered what their spins should be.

Thats my understanding of it and I dont think I'm understanding what he's trying to say entirely.

That's just about right, the particles do 'appear'to communicate with each other superluminally, but the imporatnt thing to note is that this still does not allow information to be sent superluminally.

Originally posted by jcsd No quantum entanglement does not allow information be sent superluminally, but it does mean that you can by measuring the stae of one entangled particle detrimine the state of the other entangled particle an arbitarily large distance apart.

This, I believe, is the part that throws most people. There are two statements being made here.

1) "Quantum entanglement does not allow information to be sent superluminally."

2) "You can... and determine the state of the other entangled particle."

Logically, the only way these two statements can both be true is if knowing "the state of the other entangled particle" is not "information".

Originally posted by LURCH This, I believe, is the part that throws most people. There are two statements being made here.

1) "Quantum entanglement does not allow information to be sent superluminally."

2) "You can... and determine the state of the other entangled particle."

Logically, the only way these two statements can both be true is if knowing "the state of the other entangled particle" is not "information".

Yes, but your not sending information you have to view the two particles as a single system, the only way you could if there was a way to measure whther a particle was unmeasured or not which of course is impossible as by making the measuremnt the particle would become measured.

Obviously people have tried to think of ways of sending information FTL this way, but it just can't be done, even though the two particles appear to communicate if they're measured or not superluminally.

Originally posted by cytokinesis So what you're saying is that they don't communicate superluminally, and it merely appears like they do?

Sort of, though they can be seperated by an arbitarily large distance they are still part of the same uncollapsed quantum system, therefore a measurement on one will determine the outcom of a measurement on the other, It's called the EPR paradox. I can't really reduce it anymore than to say that they appear to communicate superluminally but you cannot send information using this method.

To send communication by entanglement you need to be able to prepare one of the states with a particlar value of spin. Just measuring the value it happens to have won't allow you to communicate information.

The other thing I was unsure about was that Bell's theorem supposedly proved somewhat that superluminal communication was possible at not only a microscopic but also a macroscopic level. Maybe I'm misinterpreting what Zukav was saying, but if someone could either confirm this interpretation, or explain what part I'm missing, I'd be very appreciative.

There are a number of possible ways to explain Bell's theorem besides the existence of superluminal speeds, but only one known explanation resolves the paradox without contradiction based on first principles. Recently it was discovered by theorists that the entropy of information is proportional to the surface area. The implication is that what we are observing is a hologram, and that one of the four dimensions is an illusion.

Notably this is also what Relativity implies. Is it space or is it time? Mass or Energy? It is both and neither. Just as you cannot have a back without a front, a top without a bottom, you cannot have a particle without a wave because the two are aspects of a single unified whole we call a dimension. It is this unified whole that explains the so-called non-local effects Bell's theorem proves exist.

If you take a holographic film with a picture on it and cut it up into twenty pieces each piece contains the entire picture, just grainier. Let's say its a picture of an elephant. If you attempt to cut out just the trunk of the elephant you will get the entire elephant, just with a proportionally reduced clarity. The entropy of the information contained within the picture is proportional to the surface area.

Thus, the speed of light has been measured in laboratories as sometimes seven times faster than the usual 186,000 mph, yet the information cannot be sent any faster than usual. Instead of the light actually moving that fast what we are witnessing is apparently an illusion caused by the loss of information when we slice up nature so finely, and this could be the cause of the general randomness of Indeterminacy we observe.

One paragraph from one of Pagels'books seems to get to the point rather succinctly:

The next startling find in quantum reality shook physicists to the core. It began with a paradox formulated by Einstein and his two graduate assistants Poldosky and Rosen [usually called EPR paradox]. This was followed by thought experiments that resolved nothing until John Bell proposed a theorem, the Bell Theorem, and a real experiment to test the theorem. In a nutshell the experiment told us that the world was not locally causal! Simply put if two particles have been in contact with each and are sent out in opposite directions at the speed of light for a second, a year or a century, and then one of the particles is observed, say its polarization, the polarization of the other particle will immediately be determined: it will necessarily take the other polarization. Somehow a simultaneous change occurs in two particles separated by light years!

If we look at that metaphysically, does his theorem imply that if two objects make a connection, or interact in some way, they can keep that connection going at a great distance? What I mean is, does Bell's theorem imply ESP and telepathy between individuals is feasible? Or is my interpretation way off, yet again?

Originally posted by cytokinesis If we look at that metaphysically, does his theorem imply that if two objects make a connection, or interact in some way, they can keep that connection going at a great distance? What I mean is, does Bell's theorem imply ESP and telepathy between individuals is feasible? Or is my interpretation way off, yet again?

Beyond a doubt it does and was instrumental in Heisenburg's first formulation of the uncertainty principle. John Wheeler and Roger Penrose have championed this position for decades and their work has been inspirational. However, it remains wholly unsubstantiated. In my personal opinion, if God is playing peek-a-boo as Allan Watts put it, we mere mortals are unable to pop God's bubble.

Originally posted by wuliheron Beyond a doubt it does and was instrumental in Heisenburg's first formulation of the uncertainty principle. John Wheeler and Roger Penrose have championed this position for decades and their work has been inspirational. However, it remains wholly unsubstantiated. In my personal opinion, if God is playing peek-a-boo as Allan Watts put it, we mere mortals are unable to pop God's bubble.

What do you mean 'wholly unsubstaniated, Bell's theorum is generally accepted to be proved by the double delayed choice experiment by Alain Aspect et al.

Also this principle is the basis for quantum information being absolutely vital in quantum computing and quantum teleportation.

Bell's theorum doesn't imply anything like esp or telepathy, that's getting into the area of psychoparalllelism which is held to be incorrect by most physicists. What it does imply that the two particles are linked and can still affect each other instanteneously at distance, infact the usual way of viewing it is not as two particles but as a single quantum system.

I want to dissent from the tenor of these posts that "If you know the (whatever property) of one entangled particle, you know the (corresponding property) of the other one, no matter how far away it is".

My dissent is with the big If that starts the statement. How are you going to find out that property unless you measure it? And once you do any measurement on either particle, you destroy the entanglement.

Furthermore Bell's theorem never predicted, nor did any of the experiments ever demonstrate, anything more than an enhanced correlation between the particles.